Quadratic Functions Formula

A quadratic function is a polynomial function of degree 2, written as f(x) = ax^2 + bx + c with a ≠ 0, whose graph is a U-shaped curve called a parabola that opens upward when a > 0 or downward when a < 0.

The Formula

f(x)=ax2+bx+corf(x)=a(x−h)2+k

When to use: The path of a thrown ball — rising then falling — traces a parabola opening downward.

Quick Example

f(x)=x2−4x+3 — a parabola opening up with vertex at (2,−1).

Notation

a is the leading coefficient (determines opening direction), (h,k) is the vertex, and x=−b2a is the axis of symmetry.

What This Formula Means

A quadratic function is a polynomial function of degree 2, written as f(x)=ax2+bx+c with a≠0, whose graph is a U-shaped curve called a parabola that opens upward when a>0 or downward when a<0.

The path of a thrown ball — rising then falling — traces a parabola opening downward.

Formal View

A quadratic function f:R→R has the form f(x)=ax2+bx+c with a≠0. Its zero set is {x∈R∣ax2+bx+c=0}, with ∣zeros∣∈{0,1,2} determined by sgn⁡(b2−4ac).

Worked Examples

Example 1

easy
Find the vertex of f(x)=x2−6x+8.

Answer

(3,−1)

First step

1
The x-coordinate of the vertex is x=−b2a=−−62(1)=3.

Full solution

  1. 2
    The y-coordinate is f(3)=9−18+8=−1.
  2. 3
    The vertex is (3,−1).
For a quadratic f(x)=ax2+bx+c, the vertex formula x=−b2a gives the axis of symmetry. Plugging this x back in gives the minimum (if a>0) or maximum (if a<0) value.

Example 2

medium
Does the parabola g(x)=−2x2+4x+1 open upward or downward? Find its maximum value.

Example 3

medium
Find the vertex and axis of symmetry of f(x)=2x2−8x+3.

Common Mistakes

  • Forgetting a may be negative - then the parabola opens downward and has a maximum, not a minimum.
  • Reading the vertex from the standard form directly - use x=−b2a or convert to a(x−h)2+k.
  • Expecting one x-value per y - a horizontal line crosses a parabola twice, so two inputs can share an output.

Why This Formula Matters

Quadratics are the first non-straight function students master, introducing vertex, axis of symmetry, and the idea that one output can come from two inputs. Recognizing the ax2 term tells you to expect a curve and a turning point, not a constant rate. Recognizing it by "Is the highest power of the variable exactly 2, so the graph curves into a parabola?" — rather than by familiar numbers — is what lets a student tell it apart from linear function and quadratic formula and exponential function in a mixed problem set.

Frequently Asked Questions

What is the Quadratic Functions formula?

A quadratic function is a polynomial function of degree 2, written as f(x)=ax2+bx+c with a≠0, whose graph is a U-shaped curve called a parabola that opens upward when a>0 or downward when a<0.

How do you use the Quadratic Functions formula?

The path of a thrown ball — rising then falling — traces a parabola opening downward.

What do the symbols mean in the Quadratic Functions formula?

a is the leading coefficient (determines opening direction), (h,k) is the vertex, and x=−b2a is the axis of symmetry.

Why is the Quadratic Functions formula important in Math?

Quadratics are the first non-straight function students master, introducing vertex, axis of symmetry, and the idea that one output can come from two inputs. Recognizing the ax2 term tells you to expect a curve and a turning point, not a constant rate. Recognizing it by "Is the highest power of the variable exactly 2, so the graph curves into a parabola?" — rather than by familiar numbers — is what lets a student tell it apart from linear function and quadratic formula and exponential function in a mixed problem set.

What do students get wrong about Quadratic Functions?

The procedure for quadratic functions is the easy part; the trap is forgetting a may be negative. Asking "Is the highest power of the variable exactly 2, so the graph curves into a parabola?" first is what keeps a correct-looking calculation from being attached to the wrong concept.

What should I learn before the Quadratic Functions formula?

Before studying the Quadratic Functions formula, you should understand: linear functions, exponents.

Want the Full Guide?

This formula is covered in depth in our complete guide:

Quadratic Equations: Factoring, Completing the Square, and the Quadratic Formula →
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